Base Rep Puzzle

[Jack Brennen]

To correct my earlier post, the sequence runs as shown below (for
n=2 to 100)...  (Assuming my algorithm is correct -- it's certainly
better than my hand calculations from earlier.)

The base of each member seems to be roughly n-O(log n*log log n).
The exponent of each member seems to be roughly n*O(log n).

The sequence of bases seems to only have deltas of +1 or 0.
The sequence of exponents seems to only have positive deltas.

Note that one of the elements I gave earlier was wrong -- I
had 6^12 rather than 6^11 -- and I rewrote the first entry as
2^0 rather than 1^1, in order to preserve the property that the
exponents increase in a strictly monotonic fashion.

2^0
2^1
3^2
3^4
4^5
4^7
5^9
6^11
7^13
8^16
9^18
9^21
10^24
11^27
12^29
13^32
14^35
14^39
15^42
16^45
17^48
18^52
19^55
20^58
21^62
21^66
22^70
23^73
24^77
25^80
26^84
27^88
28^91
29^95
29^100
30^104
31^108
32^112
33^116
34^120
35^124
36^128
37^132
38^136
38^141
39^145
40^149
41^153
42^158
43^162
44^166
45^170
46^175
47^179
48^183
49^188
49^193
50^198
51^202
52^207
53^211
54^216
55^220
56^225
57^229
58^234
59^238
60^243
61^248
61^253
62^258
63^263
64^267
65^272
66^277
67^282
68^286
69^291
70^296
71^301
72^305
72^311
73^316
74^321
75^326
76^331
77^336
78^341
79^346
80^351
81^356
82^361
83^366
84^371
85^376
86^381
87^386
88^391
88^397